Your example is correct in English when you don’t know what the gender is. But — “Hi, have you met my child? Their name is Rex” makes no sense to me. “His/her” sounds more accurate to me, more consistent in the English language. But even people who are is unable or unwilling to declare a gender for themselves still refer to themselves as “I” or “me” or “my”, not “we” or “us” or “our”. It seems the height of hubris to me to demand that we refer to them in the plural.

If Miriam starts recording “they” or “them” as a way to refer to nonbinary people, I will likely adjust. Until then, I just have to live with being labeled as a homophobe or transphobe or an as-yet-to-be-defined attack label like nonbinaryphobe. That is not true, but that is the world we live in.